Soil properties show high heterogeneity at different spatial scales and their correct characterization remains a crucial challenge over large areas. The aim of the study is to quantify the impact of different types of uncertainties that arise from the unresolved soil spatial variability on simulated hydrological states and fluxes. Three perturbation methods are presented for the characterization of uncertainties in soil properties. The methods are applied on the soil map of the upper Neckar catchment (Germany), as an example. The uncertainties are propagated through the distributed mesoscale hydrological model (mHM) to assess the impact on the simulated states and fluxes. The model outputs are analysed by aggregating the results at different spatial and temporal scales. These results show that the impact of the different uncertainties introduced in the original soil map is equivalent when the simulated model outputs are analysed at the model grid resolution (i.e. 500 m). However, several differences are identified by aggregating states and fluxes at different spatial scales (by subcatchments of different sizes or coarsening the grid resolution). Streamflow is only sensitive to the perturbation of long spatial structures while distributed states and fluxes (e.g. soil moisture and groundwater recharge) are only sensitive to the local noise introduced to the original soil properties. A clear identification of the temporal and spatial scale for which finer-resolution soil information is (or is not) relevant is unlikely to be universal. However, the comparison of the impacts on the different hydrological components can be used to prioritize the model improvements in specific applications, either by collecting new measurements or by calibration and data assimilation approaches. In conclusion, the study underlines the importance of a correct characterization of uncertainty in soil properties. With that, soil maps with additional information regarding the unresolved soil spatial variability would provide strong support to hydrological modelling applications.
The prediction of mathematical environmental models is affected by uncertainty, which arises from inadequate conceptual and mathematical representations of the processes (uncertainty in model structure), inadequate and insufficient knowledge and characterization of system forcing (uncertainty in boundary conditions) and limitations in the measurements or identification of model parameters (parameter uncertainty) (Beven, 2001, 2007; Refsgaard et al., 2007; Tartakovsky et al., 2012). The need to quantify the predictive uncertainty has led to the development of probabilistic (stochastic) frameworks in many disciplines of environmental sciences and engineering (Altarejos-García et al., 2012; Baroni and Tarantola, 2014; Di Baldassarre et al., 2010; Dubois and Guyonnet, 2011; Savage et al., 2016; Seiller and Anctil, 2014). Currently rigorous quantification of uncertainty is an integral part of science-based predictions and decision support systems (Beven, 2007; Farmer and Vogel, 2016; Liu and Gupta, 2007; Montanari and Koutsoyiannis, 2012).
In hydrological studies, several sources of uncertainty have been studied ranging from atmospheric forcing (Aguilar et al., 2010; Raleigh et al., 2015; Samain and Pauwels, 2013; Vázquez and Feyen, 2003; Zhu et al., 2013) to geology structures (Comunian et al., 2016; Hansen et al., 2014; He et al., 2015; Zech et al., 2015). Among these, the uncertainty related to the soil properties has been widely analysed. Soil properties show, in fact, high heterogeneity at different spatial scales with a hierarchy of spatial structures (Burrough, 1983; Heuvelink and Webster, 2001; Vogel and Roth, 2003) and complex interactions with environmental conditions (Lin, 2010). Despite international initiatives exist to improve the current status of soil characterization (Chaney et al., 2016; Heuvelink et al., 2016; Pelletier et al., 2016; Shangguan et al., 2014), detailed information of the spatial heterogeneity of the soil properties over large areas remains a crucial challenge. For this reason, an increasing number of hydrological modelling studies aim to integrate the uncertainty in soil properties that arise from the unresolved spatial heterogeneity for a proper quantification of the uncertainty of the model results. Since soil properties play a crucial role in the entire water cycle, this topic crosses research fields from lower atmosphere (De Lannoy et al., 2014; Garrigues et al., 2015; Guillod et al., 2013; Osborne et al., 2004; Yu et al., 2014) and surface water (Anderson et al., 2006; Geza and McCray, 2008; Li et al., 2013; Livneh et al., 2015; Salazar et al., 2008) to water and solute transport to groundwater systems (Besson et al., 2011; Hennings, 2002; Yu et al., 2014).
Despite its relevance, however, relatively simple assumptions are adopted to characterize the uncertainty in soil properties and to understand its effect on the hydrological response. In several studies the uncertainty is characterized based on a relatively small number of scenarios (Baroni et al., 2010; Christiaens and Feyen, 2001; Guber et al., 2009; Herbst et al., 2006; Hohenbrink and Lischeid, 2015; Islam et al., 2006; Mirus, 2015; Moeys et al., 2012) or by simple random noise (i.e. variance) added to the original soil properties (Arnone et al., 2016; Chaney et al., 2015; Deng et al., 2009; Garrigues et al., 2015; Han et al., 2014; Loosvelt et al., 2011). Other studies explicitly integrate the complex heterogeneity of the subsurface and the uncertainty in the soil properties is characterized based on spatial correlated random fields, i.e. specifying variance and correlation length (Binley et al., 1989; Fan et al., 2016; Fiori and Russo, 2007; Merz and Plate, 1997; Meyerhoff and Maxwell, 2011). Moreover, many of the above-mentioned studies focused on the effect of the uncertainty in soil properties on a selected hydrologic variable at specific temporal and spatial scales, e.g. rainfall–runoff events (e.g. Arnone et al., 2016; Fan et al., 2016), simulated evapotranspiration (e.g. Garrigues et al., 2015), soil moisture distributions (e.g. Liao et al., 2014) or groundwater recharge (e.g. Moeys et al., 2012). Simultaneous assessments of different hydrological components of the water balance at different spatial and temporal scales are rare. In addition, due to the different settings used in the studies, it is not possible to draw general conclusions about the role of the uncertainty in soil properties. In some cases the refined spatial information of soil properties does not contribute to a more accurate prediction (e.g. Li et al., 2013). In other studies the results showed to be very sensitive to the soil properties (e.g. Livneh et al., 2015). These controversial results foster the debate on the need (or not) for finer resolution soil maps in the different modelling applications (Baveye, 2002; Baveye and Laba, 2015; Heuvelink and Webster, 2001).
In the present study, we investigate impacts of uncertainty of soil properties on hydrological states and fluxes. Uncertainty in soil properties is characterized by three different methods that are consistent in the added noise (i.e. variance), but they differ in the perturbation of the soil spatial structure, i.e. correlation length. The first two methods were previously used in other studies (e.g. Fan et al., 2016; Han et al., 2014). The third method is developed in the present study to introduce small-scale soil variability while preserving the original spatial patterns. Therefore, we hypothesize that local responses of a hydrological system, such as evapotranspiration and soil moisture, will be strongly impacted by the uncertainty introduced at small spatial scale. However, integrated responses like the streamflow aggregate local responses over large areas. We hypothesize that this integrated response will be less impacted by soil properties uncertainty. The extent of the impact is expected to decrease with increasing the aggregation area and to disappear at a specific domain size. In such a condition, the system is stated to be spatially ergodic as the model output is not any more sensitive to the perturbation, i.e. we have the equivalence between spatial and ensemble statistics (Dagan, 1989; Rubin, 2003).
The paper is structured as follow. First, the perturbation methods used for the characterization of the uncertainty of the soil properties are presented. The specific case study is described presenting the catchment, the data used and the specific settings of the perturbation methods. The hydrological model is then introduced together with the uncertainty analysis conducted for the assessment of the effect of the uncertainty in soil properties on the simulated states and fluxes. The results are discussed in Sect. 3, focusing on the effect of the differences detected at different spatial and temporal scales. Final remarks are presented in the conclusions section.
Soil perturbation methods (random error method, RE; spatially correlated method, SC; and conditional points method, CP). The panel on the left shows the percentage of sand of a hypothetical horizontal transect trough the original soil map as an orange line. Within the transect three different soil units are observed, which leads to three different sand contents. Each row of the right panels depicts the steps for setting the perturbation methods. The blue line depicts one realization of the respective perturbation method. The detailed description of these methods can be found in Sect. 2.1. Abbreviations: var – variance, CL – correlation length.
In this section, the three statistical methods to characterize the uncertainty in soil properties are presented. A sketch for describing the methods is provided in Fig. 1, where one hypothetical horizontal transect through a soil map with three soil units characterized by different percentages of sand is shown as example.
The first method (hereafter denoted as random error method – RE) is based on the assumption that the nominal value in each soil unit is the only source of uncertainty while the spatial patterns (i.e. soil units) are considered to be correct. To fulfil this assumption, a simple Gaussian random noise is defined with zero mean and given variance (Fig. 1, step R1). Random values are sampled from the distribution and added to the nominal value of soil properties of each soil unit (Fig. 1, step R2). This approach was commonly used in several studies with the focus of understanding the effect of the soil properties in forward uncertainty analysis of model response (e.g. Deng et al., 2009) or for creating the forward ensemble in data assimilation tests (e.g. Han et al., 2014).
In the second method (hereafter denoted as spatially correlated method – SC), a similar assumption of additive random values is considered. However, it is also assumed that the uncertainty arises from the presence of smaller soil units that have not been detected in the original soil map (Hennings, 2002). To fulfil this assumption, a spatial structure (i.e. variance and correlation length – CL) is defined (Fig. 1, step S1). Based on that, a spatially correlated random field with zero mean is created (Fig. 1, step S2) and added to the original soil map (Fig. 1, step S3). Random fields are used in this approach to create variability as discussed by Goovaerts (2001) with which simulated short-range components well represent the complexity of the small-scale spatial structure. Readers interested in the details of the generation of random fields are referred to Deutsch and Journel (1998), Goovaerts (1997) and Isaaks and Srivastava (1989).
Finally, in the third approach (hereafter denoted as conditional points method – CP), it is assumed that the nominal value of the original soil units represents some point locations within this unit, but their positions are unknown. The uncertainty arises from the spatial variability within these point locations that is not resolved in the original soil map. To fulfil this assumption, points are randomly distributed over the soil map and the soil properties are associated with each position (Fig. 1, step S1). These values are used to calculate the spatial structure, i.e. the empirical variogram (Fig. 1, step S2). A variogram model is fitted and a conditional random field is created using the sampled locations as conditional points (Fig. 1, step S3). It has to be noted that the CP method has some similarity with the pilot points approach used for the calibration of hydrogeological models (Carrera et al., 2005). The main difference is the use in this method of new points at each iteration; i.e. the points are located in different positions for each created conditional random field.
It is noteworthy that additional statistical methods for the analysis of soil map are presented in the literature (Goovaerts, 2011; Heuvelink et al., 2016; Kempen et al., 2009; Minasny and McBratney, 2016; Odgers et al., 2014). However, the aim of these methods is to downscale/disaggregate the information available in the original soil map and not to characterize its uncertainty. For this reason, these statistical methods are based on environmental co-variates (i.e. environmental variables that co-vary with soil variability) known at higher resolution (i.e. digital elevation model or land use) and they require relative good knowledge of the soil formation and the specific settings to adopt (Kerry et al., 2012; Nauman and Thompson, 2014; Subburayalu et al., 2014; Du et al., 2015). On the contrary, the three methods selected and developed in the present study represent relative simple approaches only based on the information available in the original soil map. They can be applied for the characterization of any type of soil properties (texture, saturated hydraulic conductivity, soil depth etc.) and they reflect different assumptions regarding the uncertainties in the soil properties. For this reason, they can be tuned to characterize uncertainty for soil maps of any scales and they can be easily used with any modelling studies (e.g. sensitivity analysis or data assimilation). Combinations of the methods can also be considered when needed; i.e. soil maps affected by different types of uncertainties.
Location of the upper Neckar catchment within Germany. The positions of the 36 gauging stations (red points) used for defining the subcatchments, the transect (dashed black line) and the two grid cells analysed (green points A and B) are depicted on the map.
The numerical experiments are conducted in the upper Neckar catchment
(Fig. 2) that was extensively investigated in
previous hydrological studies (Kumar et al.,
2010; Samaniego et al., 2010a, b; Wöhling et al., 2013b). This
catchment is located in the central uplands of Germany and comprises a
catchment area of approximately 4000 km
Soil maps of sand (%), clay (%) and bulk
density (g cm
Observed meteorological data, i.e. precipitation as well as minimum,
maximum and average daily temperature, were provided by the German
Meteorological Service (
In this section, the specific settings of each statistical perturbation method used for the characterization of the soil properties are described. The three methods are used independently to generate three different ensembles to identify the impact of the different uncertainties introduced in the original soil map on simulated states and fluxes.
Parameter settings for each perturbation method (random error, spatially correlated and conditional points). Variogram models used for the spatially correlated and conditional points methods are showed in the Supplement (Figs. S2 and S3, respectively).
Considering the random error method (see Fig. 1),
a Gaussian random additive noise is used with standard deviation
7 % and 0.07 g cm
For the spatially correlated method (see Fig. 1),
the parameters for the variogram and co-variogram models are selected to be
consistent with the perturbation introduced in the random error method
(Table 1). In particular, exponential variogram
models are prescribed with the same effective noises used in the random
error method (i.e. standard deviation 7 % and 0.07 g cm
Finally, considering the conditional points method (see Fig. 1), tests are conducted to identify the
density of the conditional points within the soil map. One sample at every 3 km
In each method, the perturbed values are forced to a realistic range, i.e. texture values between 0 and 100 % and the sum of textural fractions equal 100 %. Therefore, it has to be noted that these constrains (i) could modify the Gaussian noise introduced and (ii) could lower the uncertainty in areas of the basin where the actual values are close to the bounds. These constrains did not affect the spatial patterns of the generated soil maps of the present study due to the relative small perturbation introduced and the presence of limited areas with extreme texture conditions. However, attention has to be paid in cases where these features are more relevant.
For each method, an ensemble of 100 realizations is created to characterize the uncertainty in soil properties. The analysis is conducted with the statistical software R 3.2.x (R Core Team, 2013) using add-on packages (Pebesma, 2004). The multi-variate conditional random fields were generated with GCOSIM3D code (Gómez-Hernández and Journel, 1993).
The effect of the uncertainty in soil properties as characterized by the
three perturbation methods on hydrological states and fluxes is analysed
using the mesoscale hydrological model (mHM). The mHM
(Kumar et al., 2013; Samaniego et
al., 2010b) is an open-source, spatially distributed hydrologic model
(
The model was calibrated and validated in previous studies showing very good
capability to match streamflow measurements at catchment of different sizes
(Kumar et al., 2010,
2013; Samaniego et al., 2010b; Wöhling et al., 2013b). The same
parameterization is used for the present study. We establish the mHM over
the upper Neckar catchment at 500 m spatial resolution resulting in 16 432 grid cells.
The model run is conducted at an hourly timescale. All
simulations are conducted with a 5-year model spin-up time (1985–1989)
to minimize the effect of inappropriate initial conditions. The implications
of uncertain soil properties are evaluated showing the uncertainty in
simulated routed streamflow (SF), generated runoff at every grid cell (
Overview of the uncertainty analysis presented and discussed.
The uncertainty in simulated states and fluxes is quantified based on the
coefficient of variation (CV %) to allow for comparability between the results
obtained in the different model outputs. Assuming a generic variable
In analysis no. 1, the spatial variability of the uncertainty of the
simulated states and fluxes is presented, i.e. depending on the geographical
location within the catchment. In this case the average CV calculated for the
entire simulation period (i.e. 1 year) in each grid cell is quantified as
follows:
In analysis no. 2 (Table 2), the daily temporal
dynamic of the uncertainty obtained at each grid cell is discussed. For this
reason the CV
The uncertainty on simulated states and fluxes is further compared by
aggregating the model outputs at different resolution to identify the effect
of the spatial scale on the performance of the model as discussed by
Refsgaard et al. (2016). In particular, for use in
analysis no. 3 (Table 2), subcatchments of
different sizes are defined based on 36 gauging stations located within the
catchment (see Fig. 2). The effect of the
uncertainty in soil properties to the streamflow routed to the outlet (SF)
of each subcatchment is then compared. For the other simulated model outputs
(i.e. evapotranspiration, soil moisture and groundwater recharge), the
values of each grid cell within the subcatchment are aggregated calculating
the average of simulated model output
Finally, in analysis no. 4 (Table 2), the effect of
the aggregation of states and fluxes at different resolutions is further
analysed based on the approach shown by Hansen et al. (2014) and Rasmussen et al. (2012). In this case, the
generic model output
In addition to the spatial dimension, in this study, the same procedure is
also repeated for each spatial aggregation
The four analyses described above are conducted based on the results of 100 simulations obtained with the distributed hydrological model for each perturbation methods. A total of 300 simulations, analysed in 12 cases, are discussed in the results section (Table 2).
Soil realizations obtained for the percentage of clay based on the random error method (RE, left column), spatially correlated method (SC, middle) and conditional points method (CP, right column). The top row shows one realization for each method and the transect (dashed black line). The bottom row depicts the spread of the 100 realizations by using the 5th and 95th percentile for the selected transect (grey area). The red line depicts one realization, whereas the black line shows the percentage of clay by the original soil map.
Three methods are used to perturb the values of the original soil map, i.e.
sand (%), clay (%) and bulk density (g cm
The random error (RE) method preserves the shapes of the soil units and perturbs just the nominal values. The results therefore show how the contrasts between the soil units are modified and in some cases are exaggerated. For this reason, it is noteworthy to observe that this method could create non-realistic spatial patterns since soil properties usually show smother changes in space. The results obtained based on the spatially correlated method (SC) show that the shapes of the soil units are still highly identifiable and the sharp changes between the units are still preserved. With this method, however, the random fields superimposed on the original soil map were selected with a correlation length of 3 km (see Sect. 2.3). For this reason, smaller spatial structures than the original soil units are introduced and the sharp changes in the soil properties are not uniformly distributed all over the soil unit. Finally, considering the results obtained with the conditional point (CP) method, the results show that the soil units are visible but the contrasts are completely smoothed eliminating the artefact of the original soil map. However, the spread (grey area) in this transition between soil units (polygons) is wider than the spread detected within each soil unit. The effect is due to the combination of the uncertainty introduced to the nominal value of the soil property and to the exact position of the transition between the soil units.
The spread of the realizations is quantitatively evaluated based on the
standard deviation of the ensemble. In particular,
Fig. 5a represents the probability distribution
of the standard deviation of the clay percentage calculated at every grid
cell within the catchment (i.e. 16 432 grid cells) for each method. Results
obtained based on the three methods, on average, exhibit a high consistency
in representing the uncertainty over the catchment (i.e. average standard
deviation is for all the methods 7 %). However, some differences are
detected in the distributions. The RE method shows a normal
distribution with a relatively low variability (i.e. the coefficient of
variation of the distribution is 6 %). This is the consequence of the fact
that the soil properties within the catchment are perturbed with almost the
same magnitude. Similarly, the SC method also shows a
normal distribution but with a slightly wider variability (i.e. CV
Finally, the standard deviation of the values is calculated by aggregating
the map for different subcatchments (Fig. 5b) and
at different grid resolution (Fig. 5c) based on
the analysis described in Sect. 2.5. The spreads of the realizations
obtained with the three perturbation methods are of similar magnitude
considering the finer resolution (e.g. resolution < 1 km
In this section, the spatial variability of the uncertainty of the simulated states and fluxes is presented. In this analysis (see Sect. 2.5, Table 2, uncertainty analysis no. 1), the mean coefficient of variation over time (i.e. 1 year) is calculated for each grid cell (i.e. 16 432 grid cells) and the spatial distributions obtained with the three perturbation methods are compared (Fig. 6).
Spatial variability of the uncertainty (CV) in the simulated model
outputs (
The uncertainties of all hydrological states and fluxes obtained with each perturbation method provide nearly the same magnitude and the same spatial variability, with correlation coefficients calculated between the results obtained by each method higher than 0.8. For this reason, only the spatial distribution of the CVs of the model outputs over the entire catchment obtained with the RE method is shown, as an example (Fig. 6, left). The results obtained with all the three perturbation methods are shown for the transect depicted in Fig. 4 to facilitate the visualization of the relatively small differences (Fig. 6, right).
In general, the results obtained show that, independently from the
perturbation method used, the uncertainty in the total runoff (
For a further interpretation, the spatial variability of the uncertainty in
the simulated model outputs is compared to different boundary conditions and
input properties. In particular, the correlation coefficients between the
spatial distribution of the CVs of each model outputs and the spatial
distribution of clay (%), the mean leaf area index – LAI
(m
Correlation coefficient calculated between the spatial
distributions of the uncertainty (CV) of the simulated model outputs
(
The results obtained with the three different methods are consistent between each other also in this comparison (i.e. as represented by the small error bars) showing different correlations for each model output. The uncertainty in the runoff is stronger correlated to the actual value of the soil property. This correlation can be visually identified comparing the spatial variability detected in Fig. 6 (right) and the spatial variability of the soil property shown in Fig. 4 for the same transect. The uncertainty in the actual evapotranspiration is strongly correlated to the atmospheric conditions and, to less extend, to the soil properties. Finally, the uncertainties of the soil moisture and groundwater recharge are correlated to the vegetation characteristics, with a relatively lower effect of soil properties.
To further evaluate the different correlations found for each simulated model
output, the correlation matrix between the uncertainty (CV) detected in each
model output is calculated (Table 3). On the one hand, the results show that
the uncertainties in the fluxes are positively correlated (correlation
coefficient > 0.2). This means that when the uncertainty in one
specific flux is relatively high, also other fluxes to some degree are
uncertain. On the other hand, it is interesting to note that the uncertainty
in soil moisture is highly correlated to the groundwater recharge
(correlation coefficient
Correlation matrix of the uncertainty (CV) of the model outputs (
The daily temporal variability of the uncertainty on the simulated states and
fluxes obtained at the model resolution (i.e. 500 m) is presented in this
section. In this analysis (see Sect. 2.5, Table 2, analysis no. 2), the coefficient of variation at daily time step for
each perturbation method obtained in two grid cells selected within the
catchment are compared for an illustrative purpose. The two locations A and
B are depicted in Fig. 2. The two grid cells are
characterized by (see also Supplement, Fig. S1) a remarkable
difference in the precipitation (i.e. almost 1600 and 1000 mm yr
Daily temporal variability of the uncertainty in states and fluxes
(
The results show how the uncertainty of the total runoff is relatively high
during the entire simulation period with a tendency of increasing the
uncertainty during a high flow period. The behaviour is particularly evident
in the grid cell B (i.e. correlation coefficient between CV and simulated
runoff is 0.6). In contrast, the actual evapotranspiration is close to the
potential rate for most of the simulation period and, for this reason, it is
not sensitive to changes in soil properties. As expected, the uncertainty is
only detected during summer time when soil moisture is relatively low and
the actual evapotranspiration rate decreases in comparison to the potential
evapotranspiration. This result also explains the low correlation detected
between the uncertainty in soil moisture and evapotranspiration (see
Table 3). The temporal variability
obtained for the uncertainty in soil moisture shows a more complex behaviour
depending on the grid cell considered. In grid cell A, the CV increases with
the increasing of soil moisture while it decreases in grid cell B. The
different behaviours are explained comparing the actual soil moisture
values. In grid cell A, the soil moisture values are relatively low (0.25 m
Overall, it is noteworthy to observe how the uncertainty in soil moisture is relatively constant in time while the uncertainty in the fluxes shows much stronger temporal variability. This different behaviour can be explained considering two main characteristics. On the one hand, the presence of non-linear relations between states and fluxes generates threshold behaviour for which the uncertainty in soil moisture could be limited to ranges where the fluxes are not affected. This is for instance the case when the uncertainty in soil moisture is limited to relative wet conditions (i.e. above plant stress) and for this reason it does not affect the evapotranspiration. Similarly, the uncertainty in soil moisture could be limited in relatively dry conditions and the runoff could be not affected. On the other hand, there is a tendency of compensation in the uncertainty in the model outputs for which an overestimation of the actual evapotranspiration could be related to an underestimation of the groundwater recharge (or vice versa). In these conditions the soil moisture could be still well defined without providing any indication of the degradation of the model performance. As a result, the low uncertainty in soil moisture does not represent the overall uncertainty in the model. Overall, this analysis underlines the role of the different hydrological conditions (e.g. dry or wet) for understanding the effect of the uncertainty in soil properties on the model response. Similar conclusions are supported by the use of temporal sensitivity and identifiability analysis to better capture the role of the different uncertainties in the parameters analysed (Ghasemizade et al., 2017; Guse et al., 2016; Pianosi and Wagener, 2016; Wagener et al., 2003).
The uncertainties (CV) of simulated states and fluxes are also compared by aggregating the results over subcatchments of different sizes (see Sect. 2.5, Table 2, analysis no. 3). The results obtained with the three perturbation methods are shown against the catchment size in Fig. 9.
Uncertainty, i.e. coefficient of variation (CV), of hydrological
states and fluxes at catchments with different sizes (SF is streamflow, AET is evapotranspiration,
SM is soil moisture, GWR is groundwater recharge).
Exponential curves are fitted to the data. Please note that all figures have
individual limits for the
As presented by Refsgaard et al. (2016), the
uncertainty in all the model outputs reduces with increasing catchment area.
Assuming an arbitrary threshold (i.e. CV) acceptable for a specific model
application, this analysis identifies, on the one hand, the spatial limit of
model predictive capability for the specific application. On the other hand,
it identifies the resolution above which it might become important to have a
better understanding of the soil spatial variability. This resolution is
referred to as the Representative Elementary Scale (RES) by Refsgaard et al. (2014)
and it provides a clear and simple framework for the assessment of
the performance of distributed models. However, it is interesting to note
that the three perturbation methods generated very different results and,
assuming the same arbitrary threshold for each method, different RESs are
identified. The RE method creates higher uncertainty in all
the subcatchments and even the mean of states and fluxes over the entire
catchment is uncertain (i.e. 60 km
The different results in the uncertainty in the model outputs obtained by
the use of the different perturbation methods are consistent with the
different uncertainties introduced in the soil properties
(Fig. 5b). This result supports the conclusion
that the different RESs identified are related to the underlying
correlation length (CL) scale used in each perturbation method
(Refsgaard et al., 2016). The RE method
perturbs the value of the entire soil units and it does not generate
spatially ergodic soil parameters fields, i.e. aggregated hydrological
responses still show a non-vanishing uncertainty at large catchment. The
SC method introduced correlation length of 3 km and the
effect on the uncertainty in the aggregated model output reached a
remarkable reduction (e.g. > 90 % of the uncertainty in all
the simulated states and fluxes is reduced) when the entire catchment is
considered. Finally, the CP method introduces uncertainty
only at small spatial scales while the longer spatial patterns are
preserved. For this reason the domain is ergodic already at relatively low
catchment size (i.e. 20 km
These characteristic lengths (RES vs.
CL) identified by the use of the three different soil
perturbation methods are in agreement with previous studies conducted in
surface hydrology (Binley et
al., 1989; Fan et al., 2016; Herbst et al., 2006; Merz and Plate, 1997) and
in stochastic subsurface hydrology (Dagan, 1989; Fiori and
Russo, 2007; Rubin, 2003), where a suitable value for defining ergodic
system was found to be
A similar scaling analysis is also conducted averaging states and fluxes by coarsening the grid resolution and by aggregating at different temporal scales (see Sect. 2.5, Table 2, analysis no. 4). The results obtained with the three perturbation methods are presented in Fig. 10.
Spatio-temporal uncertainty analysis by aggregating the
model results at different spatial and temporal resolutions. The three
columns refer to the results obtained by (left) random error method - RE,
(middle) spatially correlated method – SC and (right) conditional points
method – CP. The rows refers to the different model outputs (i.e.
The spatial aggregation of the model output, as represented in the
Overall, also for this spatio-temporal analysis, the results obtained with
the three perturbation methods are very different and the RES (here defined
as the spatial and temporal scale at which it might become important – or
not – to have a better understanding of the soil spatial variability)
strongly depends on the perturbation methods used. Since the three
perturbation methods reflect different uncertainties introduced in the
original soil map, the analysis emphasizes the importance of identifying the
correct approach to characterize the uncertainty for each model application
and for further model improvements. For the specific case study presented
here, it is notable how the streamflow at the catchment outlet (i.e.
spatial resolution > 32 km), which was used for calibration of the
model in previous studies (Kumar et
al., 2013; Samaniego et al., 2010b), is sensitive only to the perturbation
of long soil spatial structures introduced with the random error method. For
this reason, it could be assumed that the uncertainty in soil properties
introduced with the RE method is well compensated by the calibration and the
RE method could not represent the actual uncertainty in the specific model
application. The same could be considered for the results obtained with the
spatially correlated method, as soon as subcatchments of different sizes are
used in the calibration. In contrast, this model output is not sensitive to
the perturbations introduced at small scale (e.g. conditional points
method). On the one hand, this means that small soil variabilities are not
relevant when the model application focuses on the streamflow prediction. On
the other hand, this results underlines that it is not possible to infer
(e.g. calibrate) these small spatial soil patterns based on the streamflow
observations. For this reason, the conditional points method appears to be a
simple and effective method to preserve the general spatial pattern of the
original soil map while introducing uncertainty due to the unresolved
spatial heterogeneity within the soil units. This type of uncertainty affects
the streamflow only for small subcatchments (size < 1 km
A different behaviour is noted for the distributed hydrological states and fluxes (evapotranspiration and groundwater recharge). These variables represent in fact local conditions (i.e. spatial resolution < 1 km) and they show the same degree of uncertainty independently from the perturbation method used. This means that these localized states and fluxes can be used to infer local properties but it is not possible to use this type of observations to calibrate the values for larger areas. For this reason, the use of, e.g., remote sensing products as total water storage anomalies and evapotranspiration is an effective approach for constraining and improving local model parameterization.
In the present study, uncertainty in soil properties is characterized based
on three statistical perturbations methods. This uncertainty is propagated
applying the distributed hydrological model mHM. The uncertainty in the
simulated states and fluxes are analysed at different spatial and temporal
scales. The main conclusions are summarized as follows:
The effect of uncertainty in soil properties depends on the hydrological
model output. In particular, the uncertainty in the fluxes are relatively
positive correlated; i.e. if the uncertainty in one of the simulated flux is
high, also the other fluxes show, to some degrees, uncertainties. On the
contrary, the uncertainty in the simulated soil moisture shows a more complex
relation as its uncertainty does not always represent the overall uncertainty
in the simulated fluxes. This behaviour is explained by the non-linear
relation between states and fluxes and the occurrence of threshold conditions
in the model response. For this reason, these results support the need for
more than one variable (e.g. soil moisture and streamflow) for a proper assessment
of the overall performance of hydrological models (Rakovec et al., 2016; Zink
et al., 2017). The uncertainty in states and fluxes depends on the specific locations and on
the boundary conditions. In particular, the uncertainty in the model results
shows strong temporal and spatial variability over the catchment with
complex interactions to local environmental conditions (i.e. atmosphere,
vegetation and soil). These results highlight the role of specific model
settings (i.e. parameters and boundary conditions) for a proper
characterization of the model response and the difficulty to generalize the
result for other applications (i.e. different study areas and weather
conditions). Similar conclusions were obtained based on sensitivity analysis
conducted using hydrological models in different catchments
(e.g. Shin et al., 2013; van Griensven
et al., 2006) and they support the use of spatial and temporal diagnostic
tools for a better understanding of the input–output space
(Ghasemizade et al., 2017; Guse
et al., 2016; Pianosi and Wagener, 2016; Wagener et al., 2003). The uncertainty in states and fluxes depends on the spatio-temporal
resolution used for the analysis. In particular, the uncertainty in all the
model outputs decreases with decreasing spatial and temporal resolution.
Assuming an arbitrary threshold (e.g. CV) acceptable for a specific model
application as proposed by Refsgaard et al. (2016), this scaling analysis
identifies the Representative Elementary Scale (RES). On the one
hand, this scale represents the resolution at which the model produces
acceptable limits of predictive capability. On the other hand, it identifies
the resolution above which it might become important to have a better
understanding of the soil spatial variability. For this reason, this
analysis proves to be a simple and practical approach for the assessment of
spatially distributed models. However, in the present study the difficulties
to identify a universal RES were identifies since it depends on
locations, time and model output. For these reasons, the present study
proposes three possible extensions of the RES approach: the use of the
maximum CV, the temporal aggregation and the assessment of multi-variables. The
first extension should better capture the model performance due to the
strong spatial and temporal variability that could be present in the
uncertainty within the catchment. The second extension could be used to
emphasize the trade-off between temporal and spatial resolution of the model
application. Finally, the third extension should provide a better assessment
of the overall performance of the model. The assumptions and the methods used for the characterizations of the
uncertainty in soil properties plays a crucial role. In particular, the
above conclusions are supported by the results obtained with all the three
soil perturbation methods used in this study. However, the absolute value of
the uncertainty detected in states and fluxes at different spatial and
temporal scales strongly depends on the perturbation methods. For this
reason, the results underline the importance to properly characterize the
specific sources of uncertainty to transform a pure numerical exercise to
specific results that are able to better support the model applications. The
three methods developed and used in the present study represent three
relatively simple approaches that can be considered to account for different
types of uncertainty in a soil map. In particular, this study proposes a new
perturbation method (here called conditional points method) able to
introduce small-scale soil variability while preserving the original spatial
patterns. In this context, however, the availability of soil map with
additional information regarding not only the actual mean value within the
soil units but also information representing the unresolved variability
(variance and correlation length of the subdominant soil units) would
provide strong support to hydrological modelling applications. Finally, the analysis conducted in the present study identifies important
information to be used for possible model improvement, either by collecting
additional data regarding the soil properties or for inverse modelling and
data assimilation frameworks. In particular, integrated fluxes such as river
discharge of large catchments are shown not to be impacted by small-scale
soil variabilities (i.e. standard deviation) but only by long spatial
structures (i.e. long correlation lengths). For this reason, additional
details in the soil map do not improve the model performance on streamflow
but rather other sources of uncertainties should be considered for that
(e.g. vegetation properties). For the same reason, this integrated
observation cannot be used to infer local parameters (i.e. parameter of
finer resolutions) but only mean characteristics of the input parameters
(e.g. average soil properties over the soil units). On the contrary, local
states and fluxes proved to be very sensitive to local variation in the soil
properties (i.e. standard deviation). For this reason, a soil map with finer
resolution data is found to be an important factor for decreasing the
uncertainty in these local model outputs. For the same reason, these
simulated outputs can be used to infer local soil parameters in calibration
or data assimilation. Despite the transition between these two extreme
conditions for which the uncertainty in soil properties is (or is not)
important is quite smooth and it depends on the output considered and on the
boundary conditions, this analysis provides a strong support to prioritize
the model improvements in specific model applications. For this reason,
similar studies can be considered for comparing statistical methods to
characterize other sources of uncertainty relevant in catchment hydrology
(e.g. precipitation, vegetation parameters).
The providers of the model input (digital elevation model, atmospheric data, soil properties, etc.) are properly referenced in the paper and in the Acknowledgements. Additional information regarding the data availability can be obtained by contacting the authors.
The authors declare that they have no conflict of interest.
The study was supported by the Deutsche Forschungsgemeinschaft (DFG) under CI 26/13-1 in the framework of the research unit FOR 2131 “Data Assimilation for Improved Characterization of Fluxes across Compartmental Interfaces” and by the Helmholtz Alliance – Remote Sensing and Earth System Dynamics (HGF-EDA). We kindly acknowledge our data providers: the German Meteorological Service (DWD), the Joint Research Center of the European Commission, the European Environmental Agency, the Federal Institute for Geosciences and Natural Re- sources (BGR), the Federal Agency for Cartography and Geodesesy (BKG). The comments provided by the two anonymous reviewers were highly appreciated.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: M. Bernhardt Reviewed by: two anonymous referees